Algebra 1 review

Question

Let a and b be non -zero rational numbers and c be a rational number . Consider operations and determine whether the result would be rational or irrational

Determine if it will be rational or irrational -

 a+B    rational or irrational ? 

 a-c     rational or irrational?

 A•b     rational or irrational ?

2/b      rational or irrational ?

a•b•c     rational or irrational?

Answer 1

Addition, Subtraction, multiplication, and division are closed on rationals...

so all of these results shall be rational as long as there is no division by zero.

Proof: a=x/y b=m/n and c=w/v where v,w,x,y,m and n are integers; x and m are not zero, as given.

a+or- b = x/y +or- m/n = (xn +or- ym)/(yn) ; because multiplication, addition and subtraction are closed

operations of the set of integers, the sum (or difference) is the quotient of two integers,

which by definition makes it rational.

the proof for multiplication and division is similar and left for you.

note that 2/b = 2/1 * n/m = (2n)/m which is also rational